Curvilinear Regression Modeling Departures from the Straight Line (Curves and Interactions)

Skill Set How does polynomial regression test for quadratic and cubic trends? What are orthogonal polynomials? When can they be used? Describe an advantage of using orthogonal polynomials over simple polynomial regression. Suppose we have one IV and we analyze this IV twice, once thru linear regression and once as a categorical variable. What does the test for the difference in R-square between the two tell us? What doesn’t it tell us, that is, if the result is significant, what is left to do?

More skills Why is collinearity likely to be a problem in using polynomial regression? Describe the sequence of tests used to model curves in polynomial regression. How do you model interactions of continuous variables with regression? What is the difference between a moderator and a mediator? How do you test for the presence of each?

Linear vs. Nonlinear Models Typical linear Model: Typical nonlinear models: We don’t use models like this: Nonlinear means in the terms, not the coefficients.

Curvilinear Regression Uses Polynomial Regression to fit curves. Polynomials are formed by taking IVs to successive powers. Polynomial equation referred to by its degree, determined by highest exponent. Power terms introduce bends. Linear Quadratic Cubic

Quadratic Function Note the bend. We can fit data with ceiling effects, sensation/perception as a function of stimulus intensity, performance as a function of practice, etc.

Quadratic Function (2) (original curve) The graph shows the effect of changing the b weight for the squared (quadratic) term.

Cubic Function Note the two bends. We get a new bend for each new power term. Sharpness of bend depends on size of b weights.

Response surfaces (1) With 1 IV, the relations between X and Y are shown as a line. With linear regression and 2 IVs, the response surface relating Y to the X variables will be a plane, e.g.: Y Y = X1+2*X2 The response surface will be like a stiff sheet of paper in a cardboard box. X1 X2

Response Surface (2) Y X1 X2 The same surface from a different angle.

Response Surface (3) Y=X1+X2-.1*X1*X1. Relations between X2 and Y are linear; relations between X1 and Y are curved. Response surface is like a section of a coffee can.

Response Surface (4) Y = X1+X2-.2X1*X1-.2X2*X2 In this graph, the relation between Y and both X variables is curved. Response surface is like a parachute.

Review What is polynomial regression? How does polynomial regression allow us to include one or more bends into lines and response surfaces?